The aim of this paper is to obtain general Cwikel-type estimates and establish CLR inequalities on noncommutative tori (a.k.a. quantum tori). Noncommutative tori are arguably the most well known examples of noncommutative spaces. In particular, they naturally appear in the noncommutative geometry approach to the quantum Hall effect [1]} and to topological insulators [2]}, [3]}. In addition, noncommutative tori have been considered in the context of string theory (see, e.g., [4]}, [5]}). Noncommutative 2-tori naturally arise from actions of \(\mathbb {Z}\)  on the circle \(\mathbb {S}^1\)  by irrational rotations. More generally, a noncommutative \(n\) -tori \(n_\theta \)  is generated by unitaries \(U_1, \ldots , U_n\)  subject to the relations,
\(U_lU_j = e^{2i\pi \theta _{jl}}U_jU_l, \qquad j,l=1,\ldots , n,\) 
